Check the raw water pump, I had a crack in one of the casting lines on my old boat. Wouldn't leak sitting there, but once running the pressure would open the crack and it would leak. Other potential points could be the rudder or engine hoses.

Could be anything. My bilge pump kept going off like yours. One time it turned out to be that a ballast fitting wasn't clamped all the way down. Another time it turned out to be that one of the exhuast flanges was leaking.

If you can't see any visible leaks from any hoses, water pumps, etc, try filling up the bilge with water (with your boat on dry land) and seeing if you can spot a leak on any of the through hull fittings. It's just a process of elimination.

When you say "not the jam nut on the drive shaft" I assume you mean the shaft packing isn't leaking.

The best thing to do is to figure out what direction the water is comming from. If the pump is running that much it shouldn't be too hard.

Pump out as much water as you can, then use a sponge to get the rest, then sit and watch the area around the bilge pump. The water will either come from the stern foward or from the bow back. That will narrow down where to look for the leak.

If the leak is in the back then I would look first at the rudder post. Some arrangements have a stuffing box for the rudder similar to the prop shaft that you can simply tighten down. Others use a rubber seal, that while it can be replaced the fact that it needs to be replaced is generally and indication that the bushing for the rudder has worn or been damaged (have you hit anything?) and now the rudder is too sloppy for the seal to work.

What I couldn't tell from your post is if the leak happens when the boat is just sitting or if it only leaks while running. If it leaks while sitting then the leak has to be something below the waterline. If it only leaks while running then it could be a lot of things.

As others have said it is also possible that you have a leak in the cooling system somewhere. Can you tell if the water is warmer than the lake water? IF the water is coming from the front, and you have a heater installed, then I would inspect the heater core as they are prone to developing leaks.

Another common failure that will cause a leak is the water circulating pump on the engine (not the raw water pump). Just like on a car, these can have the bearings start to fail and then the seal fails allowing water to leak out the shaft.

A hose can crack. Exhaust manifolds can rust through and leak. I have seen the freeze plugs on the engine rust out and leak.

This one is really stupid: I once disconnected the hose from the speedometer pitot tubes and when running there was a steady (but tiny) stream of water flowing out the tube.

I once had a boat that would take on a fair amount of water up by the bow because the fit between the deck and the hull was rather wide and the caulking had all fallen out. Everytime I turned the bow wave would wash up the side of the boat and gush into the boat through this little gap.